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Ten Lectures on the Electroweak Interactions

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Ten Lectures on the Electroweak Interactions Ten Lectures on the ElectroWeak Interactions Riccardo Barbieri Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italy Abstract Elementary particle physics is the quadrant of nature whose laws can be written in a few lines with absolute precision and the greatest empirical adequacy. If this is the case, as I believe it is, it must be possible and is probably useful to introduce the students and the interested readers to the entire subject in a compact way. This is the main aim of these Lectures. Preface Elementary particle physics is the quadrant of nature whose laws can be written in a few lines with absolute precision and the greatest empirical adequacy. If this is the case, as I believe it is, it must be possible and is probably useful to introduce the students and the interested readers to the entire subject in a compact way. This is the main aim of these Lectures. The Standard Model is the reference theory for particle physics, including the fact that one often explicitly refers to Beyond the Standard Model physics. Although maybe practical, I have never liked the distinction between Standard Model and Beyond the Standard Model physics. These lectures are certainly mostly about the Standard Model, minimally extended to include neutrino masses. As such, I avoid discussing explicitly any proposal that goes beyond the Standard Model, none of which has received yet any clear experimental confirmation. Nevertheless most of the Lectures are given with an open eye to a possible evolution of the theory of the ElectroWeak Interactions. At least I try. It will probably be most useful to read and use the Lectures with the same spirit. Not unrelated to this is the fact that, while these Lectures are being written, the commissioning of the Large Hadron Collider in Geneva is close to completion. Is it not risky, then, to write lectures about the ElectroWeak Interactions precisely when the incoming experiments at the LHC may demand a strong revision of the underlying theory? Maybe yes, but I think it is nevertheless useful to take such a risk now, at least as a way to focus on the open questions that the LHC experiments might allow to answer. There are two main difficulties in trying to give a concise course on theoretical particle physics. The first one is the number of different specialized chapters that compose nowadays particle physics. Since I think that one should resist to excessive specialization, the only (not negligible) sacrifice that I try to make in compiling the list of subjects is to leave out a discussion of strong interactions. Partially this is because I consider the Lagrangian of QuantumCromoDynamics more likely to be established then the ElectroWeak sector of the Standard Model, although not to be confused with the statement that there are no important open problems in the physics of the strong interactions. The second difficulty is of technical nature, as experienced by anybody lecturing on the subject. There is a good deal of field theory that the students should know to appreciate at best a course on theoretical particle physics. Probably as a consequence of this, several excellent books on field theory exist that include a description of the Standard Model only towards the end or at least in their second part. To be able to focus as concisely as possible on physics issues, I prefer to avoid any introduction on field theory. For this reason I include a number of short Appendices that summarize the field theory knowledge that is needed for a full understanding of the content of the various Lectures. Needless to say these Appendices cannot replace a course in field theory. Again for reasons of conciseness I skip several technical details, perhaps more than usual in a pedagogical booklet. As a partial remedy, I set problems in the course of the Lectures, without solving them explicitly. A student interested in becoming able to actively work in the field of particle physics should try to solve as many of them as possible. I thank all my collaborators, in particular Guido Altarelli, George Dvali, Gian Giudice, Lawrence Hall, Antonio Masiero, Yasunori Nomura, Riccardo Rattazzi, Andrea Romanino, Slava Rychkov and Alessandro Strumia for the many interactions, discussions, corrections of errors, etc. 1 that have influenced my view of the theory of the electroweak interactions. Finally I apologize for not providing any bibliography of the literature, which would have to be very large to be complete. 2 Contents 1 From the Fermi Theory of β-decay to the minimal Gauge Lagrangian 5 1.1 TheFermitheoryofneutrondecay . ...... 5 1.2 From the Fermi Theory to the minimal gauge Lagrangian . ........... 6 2 The Lagrangian of the Standard Model, including neutrino masses 9 2.1 The global symmetries of the minimal gauge Lagrangian . ............ 9 2.2 TowardsarealisticLagrangian. ........ 10 2.3 The accidental symmetries of the Standard Model . ........... 10 3 The main predictions of the Standard Model 12 3.1 Gauge symmetry breaking and particle masses . .......... 12 3.2 Couplingstofermionsofthegaugebosons . ......... 13 3.3 TheHiggsboson ................................... 14 4 Precision tests 15 4.1 Parity violation in atomic physics . ......... 15 4.2 Leading corrections to the ρ parameter ........................ 16 4.3 SensitivitytotheHiggsmass. ....... 17 4.4 Vacuum polarization amplitudes in a general universal theory ........... 19 4.5 Current experimental constraints . ......... 21 4.6 An interlude: making it without a Higgs boson . .......... 22 5 Flavour Physics 25 5.1 The theorems of flavour physics . ...... 25 5.2 Individual lepton number conservation . .......... 25 5.3 About the unitarity of the Cabibbo-Kobaiashi-Maskawa matrix .......... 27 5.4 Calculable Flavour Changing Neutral Current processes ............... 28 5.5 Summary of calculable FCNC processes . ........ 31 6 CP violation 32 6.1 The source(s) of CP violation in the Lagrangian of the StandardModel . 32 6.2 Electricdipolemoments . ..... 33 6.3 CP violation in effective 4-fermion interactions . .............. 35 7 Basics of neutrino physics 38 7.1 The three options for neutrino masses in the Standard Model............ 38 7.2 Thephysicalparameters . ..... 39 7.3 Neutrino mass measurements from the β-decayspectrum . 40 7.4 Neutrino-less double-β decay.............................. 40 3 8 Neutrino oscillations 42 8.1 Neutrino oscillations in vacuum . ........ 42 8.2 Neutrinopropagationinmatter . ....... 43 8.3 Current determination of neutrino masses and mixings . ............. 45 9 The naturalness problem of the Fermi scale 48 9.1 The Standard Model as a prototype effective theory . ........... 48 9.2 Expanding in operators of higher dimension . .......... 48 9.3 MinimalFlavourViolation . ...... 50 9.4 The naturalness scale of the Standard Model . .......... 50 9.5 The little hierarchy problem .............................. 52 10 The main drawback of the Standard Model 53 10.1 Gauge anomalies and charge quantization. ........... 53 10.2 Theunificationway............................... .... 54 A General structure of a gauge theory 56 B Real and chiral representations of the gauge group 57 G C Spontaneous breaking of a gauge or a global symmetry 58 D Renormalizable theories and effective theories 60 E CP invariance 61 F Weyl, Dirac and Majorana neutrinos 62 G Anomalies 63 4 1 From the Fermi Theory of β-decay to the minimal Gauge Lagrangian 1.1 The Fermi theory of neutron decay In 1934 Fermi wrote the first effective Lagrangian to describe a weak interaction phenomena: nuclear β-decay. Considering neutron decay, the Fermi Lagrangian can be written, with current knowledge, as GF F = cos θC (¯pγµ(1 + αγ5)n)(¯eγµ(1 γ5)ν) (1.1) L √2 − where p, n, e, ν are the fields of the proton, neutron, electron and neutrino respectively. In units 2 where h/ = c = 1, GF is a constant of dimension of mass− , since every fermion field, as readily 3/2 seen from the free Lagrangian, has dimension of mass . The presence of the factor cos θC , close to unity, will be commented later on. Finally α is another dimensionless constant. This interaction allows to calculate both the neutron width and its angular dependence. For the total width one finds G2 ∆5 Γ= F cos2 θ (1+3α2)Φ (1.2) 60π3 C where ∆ = 1.29 MeV is the neutron-proton mass difference and Φ = 0.47 is a numerical factor that would be unity if the electron mass were neglected relative to ∆. The angular dependence of the width in the neutron rest frame is given by dΓ 1 α2 (1 + − 2 ve n) (1.3) dΩe ∝ 1+3α · where ve and n are the 3-velocities of the electron and of the neutrino respectively. [Problem 1.1.1: Prove eq.s (1.2) and (1.3) starting from eq. (1.1).] The measurements of the lifetime and of the angular distribution give 1 τ = = 885.7 0.8 sec, α = 1.2695 0.0029, (1.4) Γ ± − ± 1/2 from which one infers, using eq. (1.2), G− 250 GeV . As we shall see, this ”Fermi scale” plays F ≈ a fundamental role in the theory of the electroweak interactions. It is believed to be one of the two fundamental scales in particle physics, the other being the scale of the strong interactions or the scale of Quantum CromoDynamics. Based on the analogy with Quantum ElectroDynamics, Fermi himself, among others, conjec- tured that the interaction in eq. (1.1) could result from the exchange of a heavy charged vector boson, Wµ±, of mass mW , interacting with the current 1+ αγ5 1 γ5 + + J − =pγ ¯ n +¯νγ − e, J =(J −) (1.5) µ µ 2 µ 2 µ µ via g + int = W J − + h.c., (1.6) L √2 µ µ 5 where g is a dimensionless coupling. Since ∆ is negligible with respect to mW , the exchange of the W -boson gives indeed rise to the Fermi interaction in eq.
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